We describe the relationship between a relaxed problem (REF ) and its approximations (REF ) using the notion of epi-convergenceFor background information about epi-convergence; see for example [1]}.. The functions \(\psi ^\nu :\mathbb {R}^n\rightarrow \overline{\mathbb {R}}\)  are said to epi-converge to \(\psi :\mathbb {R}^n\rightarrow \overline{\mathbb {R}}\)  when
\(&\forall x^\nu \rightarrow x, ~\mathop {\rm liminf}\nolimits \psi ^\nu (x^\nu ) \ge \psi (x)\\&\forall x, ~\exists x^\nu \rightarrow x \mbox{ with } \mathop {\rm limsup}\nolimits \psi ^\nu (x^\nu ) \le \psi (x).\) 
